In many dynamical systems it is possible to choose the coordinates of the system so that the volume (really a ν-dimensional volume) in phase space is invariant. This happens for mechanical systems derived from Newton's laws as long as the coordinates are the position and the momentum and the volume is measured in units of (position) × (momentum). The flow takes points of a subset A into the points Φ t(A) and invariance of the phase space means that

\mathrm{vol} (A) = \mathrm{vol} ( \Phi^t(A) ) \,.

In the Hamiltonian formalism, given a coordinate it is possible to derive the appropriate (generalized) momentum such that the associated volume is preserved by the flow. The volume is said to be computed by the Liouville measure.

In a Hamiltonian system not all possible configurations of position and momentum can be reached from an initial condition. Because of energy conservation, only the states with the same energy as the initial condition are accessible. The states with the same energy form an energy shell Ω, a sub-manifold of the phase space. The volume of the energy shell, computed using the Liouville measure, is preserved under evolution.

For systems where the volume is preserved by the flow, Poincaré discovered the recurrence theorem: Assume the phase space has a finite Liouville volume and let F be a phase space volume-preserving map and A a subset of the phase space. Then almost every point of A returns to A infinitely often. The Poincaré recurrence theorem was used by Zermelo to object to Boltzmann's derivation of the increase in entropy in a dynamical system of colliding atoms.

One of the questions raised by Boltzmann's work was the possible equality between time averages and space averages, what he called the ergodic hypothesis. The hypothesis states that the length of time a typical trajectory spends in a region A is vol(A)/vol(Ω).

The ergodic hypothesis turned out not to be the essential property needed for the development of statistical mechanics and a series of other ergodic-like properties were introduced to capture the relevant aspects of physical systems. Koopman approached the study of ergodic systems by the use of functional analysis. An observable a is a function that to each point of the phase space associates a number (say instantaneous pressure, or average height). The value of an observable can be computed at another time by using the evolution function φ t. This introduces an operator U t, the transfer operator,

(U^t a)(x) = a(\Phi^{-t}(x)) \,.

By studying the spectral properties of the linear operator U it becomes possible to classify the ergodic properties of Φ t. In using the Koopman approach of considering the action of the flow on an observable function, the finite-dimensional nonlinear problem involving Φ t gets mapped into an infinite-dimensional linear problem involving U.

The Liouville measure restricted to the energy surface Ω is the basis for the averages computed in equilibrium statistical mechanics. An average in time along a trajectory is equivalent to an average in space computed with the Boltzmann factor exp(−βH). This idea has been generalized by Sinai, Bowen, and Ruelle (SRB) to a larger class of dynamical systems that includes dissipative systems. SRB measures replace the Boltzmann factor and they are defined on attractors of chaotic systems.

Nonlinear dynamical systems and chaos

Main article: Chaos theory

Simple nonlinear dynamical systems and even piecewise linear systems can exhibit a completely unpredictable behavior, which might seem to be random. (Remember that we are speaking of completely deterministic systems!). This seemingly unpredictable behavior has been called chaos. Hyperbolic systems are precisely defined dynamical systems that exhibit the properties ascribed to chaotic systems. In hyperbolic systems the tangent space perpendicular to a trajectory can be well separated into two parts: one with the points that converge towards the orbit (the stable manifold) and another of the points that diverge from the orbit (the unstable manifold).

This branch of mathematics deals with the long-term qualitative behavior of dynamical systems. Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible attractors?" or "Does the long-term behavior of the system depend on its initial condition?"

Note that the chaotic behavior of complicated systems is not the issue. Meteorology has been known for years to involve complicated—even chaotic—behavior. Chaos theory has been so surprising because chaos can be found within almost trivial systems. The logistic map is only a second-degree polynomial; the horseshoe map is piecewise linear.

Geometrical definition

A dynamical system is the tuple \langle \mathcal{M}, f , \mathcal{T}\rangle , with \mathcal{M} a manifold (locally a Banach space or Euclidean space), \mathcal{T} the domain for time (non-negative reals, the integers, ...) and f an evolution rule t→f t (with t\in\mathcal{T}) such that f t is a diffeomorphism of the manifold to itself. So, f is a mapping of the time-domain \mathcal{T} into the space of diffeomorphisms of the manifold to itself. In other terms, f(t) is a diffeomorphism, for every time t in the domain \mathcal{T} .

Measure theoretical definition

See main article Measure-preserving dynamical system.

A dynamical system may be defined formally, as a measure-preserving transformation of a sigma-algebra, the quadruplet (X,Σ,μ,τ). Here, X is a set, and Σ is a sigma-algebra on X, so that the pair (X,Σ) is a measurable space. μ is a finite measure on the sigma-algebra, so that the triplet (X,Σ,μ) is a probability space. A map \tau:X\to X is said to be Σ-measurable if and only if, for every \sigma \in \Sigma, one has \tau^{-1}\sigma \in \Sigma. A map τ is said to preserve the measure if and only if, for every \sigma \in \Sigma, one has μ(τ − 1σ) = μ(σ). Combining the above, a map τ is said to be a measure-preserving transformation of X , if it is a map from X to itself, it is Σ-measurable, and is measure-preserving. The quadruple (X,Σ,μ,τ), for such a τ, is then defined to be a dynamical system.

The map τ embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the iterates \tau^n=\tau \circ \tau \circ \ldots\circ\tau for integer n are studied. For continuous dynamical systems, the map τ is understood to be a finite time evolution map and the construction is more complicated.

\mathrm{vol} (A) = \mathrm{vol} ( \Phi^t(A) ) \,.

In the Hamiltonian formalism, given a coordinate it is possible to derive the appropriate (generalized) momentum such that the associated volume is preserved by the flow. The volume is said to be computed by the Liouville measure.

In a Hamiltonian system not all possible configurations of position and momentum can be reached from an initial condition. Because of energy conservation, only the states with the same energy as the initial condition are accessible. The states with the same energy form an energy shell Ω, a sub-manifold of the phase space. The volume of the energy shell, computed using the Liouville measure, is preserved under evolution.

For systems where the volume is preserved by the flow, Poincaré discovered the recurrence theorem: Assume the phase space has a finite Liouville volume and let F be a phase space volume-preserving map and A a subset of the phase space. Then almost every point of A returns to A infinitely often. The Poincaré recurrence theorem was used by Zermelo to object to Boltzmann's derivation of the increase in entropy in a dynamical system of colliding atoms.

One of the questions raised by Boltzmann's work was the possible equality between time averages and space averages, what he called the ergodic hypothesis. The hypothesis states that the length of time a typical trajectory spends in a region A is vol(A)/vol(Ω).

The ergodic hypothesis turned out not to be the essential property needed for the development of statistical mechanics and a series of other ergodic-like properties were introduced to capture the relevant aspects of physical systems. Koopman approached the study of ergodic systems by the use of functional analysis. An observable a is a function that to each point of the phase space associates a number (say instantaneous pressure, or average height). The value of an observable can be computed at another time by using the evolution function φ t. This introduces an operator U t, the transfer operator,

(U^t a)(x) = a(\Phi^{-t}(x)) \,.

By studying the spectral properties of the linear operator U it becomes possible to classify the ergodic properties of Φ t. In using the Koopman approach of considering the action of the flow on an observable function, the finite-dimensional nonlinear problem involving Φ t gets mapped into an infinite-dimensional linear problem involving U.

The Liouville measure restricted to the energy surface Ω is the basis for the averages computed in equilibrium statistical mechanics. An average in time along a trajectory is equivalent to an average in space computed with the Boltzmann factor exp(−βH). This idea has been generalized by Sinai, Bowen, and Ruelle (SRB) to a larger class of dynamical systems that includes dissipative systems. SRB measures replace the Boltzmann factor and they are defined on attractors of chaotic systems.

Nonlinear dynamical systems and chaos

Main article: Chaos theory

Simple nonlinear dynamical systems and even piecewise linear systems can exhibit a completely unpredictable behavior, which might seem to be random. (Remember that we are speaking of completely deterministic systems!). This seemingly unpredictable behavior has been called chaos. Hyperbolic systems are precisely defined dynamical systems that exhibit the properties ascribed to chaotic systems. In hyperbolic systems the tangent space perpendicular to a trajectory can be well separated into two parts: one with the points that converge towards the orbit (the stable manifold) and another of the points that diverge from the orbit (the unstable manifold).

This branch of mathematics deals with the long-term qualitative behavior of dynamical systems. Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible attractors?" or "Does the long-term behavior of the system depend on its initial condition?"

Note that the chaotic behavior of complicated systems is not the issue. Meteorology has been known for years to involve complicated—even chaotic—behavior. Chaos theory has been so surprising because chaos can be found within almost trivial systems. The logistic map is only a second-degree polynomial; the horseshoe map is piecewise linear.

Geometrical definition

A dynamical system is the tuple \langle \mathcal{M}, f , \mathcal{T}\rangle , with \mathcal{M} a manifold (locally a Banach space or Euclidean space), \mathcal{T} the domain for time (non-negative reals, the integers, ...) and f an evolution rule t→f t (with t\in\mathcal{T}) such that f t is a diffeomorphism of the manifold to itself. So, f is a mapping of the time-domain \mathcal{T} into the space of diffeomorphisms of the manifold to itself. In other terms, f(t) is a diffeomorphism, for every time t in the domain \mathcal{T} .

Measure theoretical definition

See main article Measure-preserving dynamical system.

A dynamical system may be defined formally, as a measure-preserving transformation of a sigma-algebra, the quadruplet (X,Σ,μ,τ). Here, X is a set, and Σ is a sigma-algebra on X, so that the pair (X,Σ) is a measurable space. μ is a finite measure on the sigma-algebra, so that the triplet (X,Σ,μ) is a probability space. A map \tau:X\to X is said to be Σ-measurable if and only if, for every \sigma \in \Sigma, one has \tau^{-1}\sigma \in \Sigma. A map τ is said to preserve the measure if and only if, for every \sigma \in \Sigma, one has μ(τ − 1σ) = μ(σ). Combining the above, a map τ is said to be a measure-preserving transformation of X , if it is a map from X to itself, it is Σ-measurable, and is measure-preserving. The quadruple (X,Σ,μ,τ), for such a τ, is then defined to be a dynamical system.

The map τ embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the iterates \tau^n=\tau \circ \tau \circ \ldots\circ\tau for integer n are studied. For continuous dynamical systems, the map τ is understood to be a finite time evolution map and the construction is more complicated.

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